離散數學 第二章小考 姓名: ____________________ 學號:_________________
1. (18%) Let A, B, C be finite sets. Find each of the following: Mark the following statement TRUE or FALSE
Note: AÅB,$is$the$set$containing$those$elements$in$either$A$or$B,$but$not$in$both$A$an$B.$(每小題直接圈選T/F即可)$
(a)$$T$$F$$ |A∪B|=|A|+|B|–|A∩B|$
(b)$$T$$F$$ P(A∪B)="P(A)∪P(B)$
(c)$$T$$F$$ A$–$(B∪C)=$(A$–B)$–$C)$
(d)$$T$$F$$ If$A∪C=$B∪C,$then$A=B$
(e)$$T$$F$$ If$A∩C=$B∩C,$then$A=B$
(f)$$T$$F$$ IF$A$–$C=B–C,$then$A=B$
(g)$$T$$F$$ If$A={a,c,c,e,e,e}$and$B={e,c,a},$then$A=B$
(h)$$T$$F$$ If$AÅC=$BÅC,$then$A=B$
(i)$$T$$F$$ A∩(BÅC)=$(A∩B)Å($A∩C)$
(j)$$T$$F$$ Ø$Î$Ø$
(k)$$T$$F$$ Ø$Î${Ø}$
(l)$$T$$F$$ Ø$Ì$Ø$
(m)$$T$$F$$ Ø$Í{Ø}$
(n)$$T$$F$$ Ø$Í{{Ø}}$
(o)$$T$$F$$ {Ø}$Î${Ø,{{$Ø$}}}$
(p)$$T$$F$$ {{Ø},{Ø}}$Í{Ø,{Ø}}$
(q)$$T$$F$$ {{Ø}}$Ì${{Ø},{Ø}}$
(r)$$T$$F$$ |$Ø$|=1$
Ans:$T$F$T$F$F$F$T$T$T$F$T$F$T$T$F$T$F$F$
2. (10%)$Let!f:$A®B$and$g:$B®C$be$functions.$Mark$the$following$statement$TRUE$or$FALSE.$(直接圈選T/F即可)$
(a)$$T$$F$$ If$g
。
f#is$one-to-one,$so$is$f.$
(b)$$T$$F$$ If$g
。
f#is$one-to-one,$so$is$g.$
(c)$$T$$F$$ If$g
。
f#is$onto,$so$is$f.$
(d)$$T$$F$$ If$g
。
f#is$onto,$so$is$g.$
(e)$$T$$F$$ If$g
。
f#is$bijection,$so$are$g#and#f.$
Ans:$T$F$F$T$F$(參考圖如下)$
$
3. (16%)$Let$A$and$B$be$arbitrary$infinitely$countable$sets;$C$ and$D$be$arbitrary$infinitely$uncountable$sets.$Mark$
the$following$statement$TRUE$or$FALSE.$(直接圈選T/F即可)$
(a)$$T$$F$$ A–B$can$be$Ø.$
(b)$$T$$F$$ A–B$can$be$finite.$
(c)$$T$$F$$ A–B$can$be$infinitely$countable.$
(d)$$T$$F$$ A–B$can$be$uncountable.$
(e)$$T$$F$$ C$–D$can$be$Ø$
(f)$$T$$F$$ C$–D$can$be$finite$
(g)$$T$$F$$ C-D$can$be$infinitely$countable$
(h)$$T$$F$$ C-D$can$be$uncountable$
Ans:$T$T$T$F$T$T$T$T,$[其中(g)的例子:$A=R,$B=R-Z+$,則A-B=$Z+]$
4. (8%)$Let!f,$g:$R®R,$where$g(x)=1-x+x2$and$f(x)=ax+b.$If$(g
。
f)(x)=9x2-9x+3,$determine$a,$b.$
Ans:$a=3,$b=-1$ 或$a=-3,$b=2$
5. (5%) Draw the graph of the function f(x)=é0.4xù from R to R.
Ans: (略)
A
B
C
f
g
g
。
f
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分6b 3⼆9byAmi9
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3，52
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